If A and B are two independent events and \( P(A)=\displaystyle\frac{1}{2},\ P(B)=\displaystyle\frac{1}{3} \), then
Match List-I with List-II
| List-I | List-II |
| (A) \( P(A\cup B) \) | (I) \( \displaystyle\frac{5}{6} \) |
| (B) \( P(A\cap\overline{B}) \) | (II) \( \displaystyle\frac{1}{6} \) |
| (C) \( P(\overline{A}\cap B) \) | (III) \( \displaystyle\frac{2}{3} \) |
| (D) \( P(\overline{A\cap B}) \) | (IV) \( \displaystyle\frac{1}{3} \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) - (III), (B) - (IV), (C) - (II), (D) - (I)
For independent events, the complements are also independent, and
\( \displaystyle P(A\cap B)=\frac{1}{2}\times\frac{1}{3}=\frac{1}{6} \)
(A) \( P(A\cup B) \)
This matches (III) \( \frac{2}{3} \).
\( \displaystyle \frac{1}{2}+\frac{1}{3}-\frac{1}{6}=\frac{2}{3} \)
(B) \( P(A\cap\overline{B}) \)
This matches (IV) \( \frac{1}{3} \).
\( \displaystyle P(A)\,P(\overline{B})=\frac{1}{2}\times\frac{2}{3}=\frac{1}{3} \)
(C) \( P(\overline{A}\cap B) \)
This matches (II) \( \frac{1}{6} \).
\( \displaystyle P(\overline{A})\,P(B)=\frac{1}{2}\times\frac{1}{3}=\frac{1}{6} \)
(D) \( P(\overline{A\cap B}) \)
This matches (I) \( \frac{5}{6} \).
\( \displaystyle 1-P(A\cap B)=1-\frac{1}{6}=\frac{5}{6} \)
So the matching is (A) - (III), (B) - (IV), (C) - (II), (D) - (I).